A spherical balloon is inflated to a diameter of 20.0 cm. Assume that the gas in the balloon is of atmospheric pressure (101.3 kPa) and is at a temperature of 20.0 ¡ãC. It is then taken by a diver 15.0 m under the sea. The temperature of the seawater at this depth is 16.0 ¡ãC . Density of seawater: 1030. kg m¨C3. The gauge pressure of the air in the balloon at this depth is 27300Pa.
Assuming the gas in the balloon is in thermal equilibrium with seawater, what is the volume of the balloon now?
Thanks in advance !!!!!
To find the volume of the balloon after being taken 15.0 m under the sea, we need to consider the ideal gas law and the effect of pressure and temperature changes.
The ideal gas law equation is given by:
PV = nRT,
where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature in Kelvin.
Since we have the pressure, temperature, and volume of the balloon before it is taken under the sea, we can use this to find the initial number of moles of gas in the balloon.
First, let's convert the atmospheric pressure from kilopascals to pascals to match the units:
101.3 kPa = 101300 Pa.
Also, we need to convert the temperature from Celsius to Kelvin:
20.0°C + 273.15 = 293.15 K.
Now, we can rearrange the ideal gas law equation to solve for the initial number of moles (n):
n = PV / RT.
Substituting the given values, we get:
n = (101300 Pa) * (V_initial) / ((8.314 J/(mol*K)) * (293.15 K)),
where V_initial is the initial volume of the balloon.
Next, let's consider the situation after the balloon is taken 15.0 m under the sea. We are given the gauge pressure at this depth, which is the difference between the absolute pressure and the atmospheric pressure. Therefore, the absolute pressure is:
P_absolute = gauge pressure + atmospheric pressure.
Substituting the values, we have:
P_absolute = 27300 Pa + 101300 Pa = 128600 Pa.
Now, we need to determine the temperature of the seawater at the depth of 15.0 m, which is given as 16.0°C. Convert it to Kelvin:
16.0°C + 273.15 = 289.15 K.
Since the gas in the balloon is in thermal equilibrium with the seawater, we assume that their temperatures are the same.
Now, we can find the final volume of the balloon using the ideal gas law equation:
V_final = (n * R * T_final) / P_absolute,
where T_final is the final temperature, which we assumed to be the temperature of the seawater at the depth of 15.0 m (289.15 K).
Substituting the values, we can calculate the volume (V_final) of the balloon.